Abstract
One of the basic techniques in every mathematician’s toolkit is the Taylor series representation of functions. It is of such fundamental importance and it is so well understood that its use is often a first choice in numerical analysis. This faith has not, unfortunately, been transferred to the design of computer algorithms.
Approximation by use of Taylor series methods is inherently partly a symbolic process and partly numeric. This aspect has often, with reason, been regarded as a major hindrance in algorithm design. Whilst attempts have been made in the past to build a consistent set of programs for the symbolic and numeric paradigms, these have been necessarily multi-stage processes.
Using current technology it has at last become possible to integrate these two concepts and build an automatic adaptive symbolic-numeric algorithm within a uniform framework which can hide the internal workings behind a modern interface.
The project “Composite Computing Methods Integrating Symbolic, Numeric and Graphical Packages for Research Engineers” is funded by the UK Govt. Joint Information Systems Committee under their Technology Applications Programme JTAP 5/11
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References
Barton, D. On taylor series and stiff equations. ACM Trans. Math. Softw. 6, 3(Sept. 1980), 281–294.
Barton, D., Willers, I. M., and Zahar, R. M. V. The automatic solution of systems of ordinary differential equations by the method of taylor series. Computer Journal 14, 3 (1971).
Barton, D., Willers, I. M., and Zahar, R. M. V. Taylor series methods for ordinary differential equations — an evaluation. In Rice [14], pp. 369–390.
Broughan, K. A., Keady, G., Robb, T., Richardson, M. G., and Dewar, M. C. Some symbolic computing links to the NAG numeric library. SIGSAM Bulletin 25 (July 1991), 28–37.
Corliss, G. F., and Chang, Y. F. Solving ordinary differential equations using taylor series. ACM Trans. Math. Softw. 8, 2 (June 1982), 114–144.
Dewar, M. C. Manipulating fortran code in AXIOM and the AXIOM-NAG link. In Workshop on Symbolic and Numeric Computation (1993)(Helsinki, 1994), H. Apiola, M. Laine, and E. Valkeila, Eds., pp. 1–12. Research Report BIO, Rolf Nevanlinna Institute, Helsinki
Dupée, B. J., and Davenport, J. H. An intelligent interface to numerical routines. In DISCO’96: Design and Implementation of Symbolic Computation Systems(Karlsruhe, 1996), J. Calmet and J. Limongelli, Eds., vol. 1128 of Lecture Notes in Computer Science, Springer Verlag, Berlin, pp. 252–262.
Heck, A. Introduction to Maple, 2nd ed. Springer Verlag, New York, 1996.
Jenks, R. D., and Sutor, R. S. AXIOM: The Scientific Computation System. Springer-Verlag, New York, 1992.
Keady, G., and Nolan, G. Production of argument subprograms in the AXIOM-NAG link: examples involving nonlinear systems. In Workshop on Symbolic and Numeric Computation, May 1993(Helsinki, 1994), H. Apiola, M. Laine, and E. Valkeila, Eds., pp. 13–32. Research Report BIO, Rolf Nevanlinna Institute, Helsinki.
Norman, A. C. Taylor Users Manual. Computing Laboratory, University of Cambridge, UK, 1973.
Norman, A. C. Computing with formal power series. ACM Trans. Math. Softw. 1 (1975), 346–356.
Norman, A. C. Expanding the solutions of implicit sets of ordinary differential equations. Comp. J 19 (1976), 63–68.
Rice, J. R., Ed. Mathematical Software. Academic Press, New York,1971
Wolfram, S. The Mathematica Book, 3rd ed. CUP, Cambridge, 1996.
Zwillinger, D. Handbook of Differential Equations, 2nd ed. Academic Press, San Diego, CA, 1989.
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Dupée, B.J., Davenport, J.H. (1999). An Automatic Symbolic-Numeric Taylor Series ODE Solver. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing CASC’99. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60218-4_3
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